The document summarizes an experiment on speed control of a DCMCT motor unit using proportional-integral (PI) control. There were four experiments: 1) proportional control, 2) integral control, 3) PI control with set point bsp=0, and 4) PI control with bsp=1. The optimal results were obtained in experiment 3 when manually tuning the proportional gain kp and integral gain ki to values of 2.25 and 0.19 respectively, achieving no steady state error, no overshoot, and a settling time less than 0.25 seconds. The experiments helped understand how changing control parameters affects the speed response and validated theoretical models of the different control methods.

1. i
The Analysis of a Speed Control Using
P-I Control
Control Systems ME 451
Dominic Waldorf
Section 006 Group C
Wednesday 7:00 PM
Dr. Jongeun Choi and TA Nilay Kant
March 22, 2016

2. ii
Abstract:
Speed control using P-I control is an important method in the understanding of
control system. The experiment assisted in proving the theoretical pure proportional
control, integral control, and proportional and integral control values to the experimental
values. The task was accomplished by deriving an equation from the block diagram of
the system. Once the transfer function was calculated, the equation was manipulated to
give the kpc, kic, Ts, ki, and kp values for each of the different methods of response. The
experiment also assisted in understanding what happens when those values are changed.
The experiment was performed using a DCMCT motor unit and the Labview
software on the desktop. There were four different experiments performed. For each
experiment the values were adjusted, giving different response for each experiment.
The optimal results were given when manually tuning the parameters to improve
the settling time. The goal is to choose the kp and ki values so there is no steady state
error, there is no overshoot, and the 2% settling time is less than or equal to .25 seconds.
This was done during the proportional and integral control when bsp was set to zero. The
old values were ki=2.25, kp=.19, and Ts=.28. The new/optimal values were ki=1.6356,
kp=.148, and Ts=.66.

3. iii
Table of Contents
Nomenclature Listing.....................................................................................................1
Introduction.....................................................................................................................2
Theory and Analysis.......................................................................................................2
Experimental Equipment and Procedure ..................................................................4
Results............................................................................................................................... 6
Discussion.......................................................................................................................11
Conclusion ......................................................................................................................12
Reference........................................................................................................................13

4. 1
Nomenclature Listing
bsp set-point constant
K Gain
kp Proportional Gain Contstant
kpc
Critically Damped Proportional gain
constant
kpu Gain required for marginal instability
ki integral gain constant
kic critically damped integral gain constant
Ts settling time
Tu period of oscillation at marginal instability
tc output low pass filter time constant
τ time constant
Table 1: variable definition

5. 2
Introduction
Speed control using a P-I control is a very useful tool for understanding the
operations of a DCMCT motor unit control system. The experiment performed is very
important to for Mechanical Engineers to understand because it helps give the engineer a
firm grasp on how to manipulate parameters to achieve the desired results. These results
can help better tune controllers such as cruise control in a car or even an airplane. If the
controller is better tuned and able to adjust to different external interactions, it will
improve the safety of the vehicle. Not only will it improve safety, but it will also
improve the performance of the engine. The goal of the experiment is to have no steady
state error, no overshoot, and the 2% settling time less than or equal to .25 seconds.
Theory and Analysis
The PI control to the DC motor plant was used in the experiment. A block
diagram modeled the system. The block diagram was used to derive the transfer function
which is as follow:
Figure 1: Block Diagram of Transfer Function

6. 3
𝐻( 𝑠) =
𝐾∗𝑘𝑝∗𝑏𝑠𝑝∗𝑠+𝐾∗𝑘𝑖
𝜏∗𝑡𝑐∗𝑠3 +( 𝑡𝑐+𝜏)∗𝑠2 +( 𝐾∗𝑘𝑝+1)∗𝑠+𝐾∗𝑘𝑖
(1)
After each of the control methods were given certain parameters based on the desired
outcome. These values are the theoretical values for the experiment. These equations
were derived in the pre-lab on the experiment.
Proportional Control:
(ki = 0)
𝑘𝑝𝑐 = (
1
4
∗
(( 𝑡𝑐+𝜏)2
𝑡𝑐∗𝜏
− 1) ∗
1
𝐾
(2)
Integral Control:
(kp=0)
𝑘𝑖𝑐 =
1
4𝐾∗( 𝑡𝑐+𝜏)
(3)
𝑇𝑠 = 8 ∗ ( 𝑡𝑐 + 𝜏) (4)
Proportional and Integral Control:
(bsp = 0)
𝑇𝑠 = .25 (5)
𝑘𝑖 = 256 ∗
𝑡𝑐+𝜏
𝐾
(6)
𝑘𝑝 =
1
𝑘
∗ (2 ∗ ( 𝑡𝑐 + 𝜏) ∗ √
𝐾∗𝑘𝑖
𝑡𝑐+𝜏
− 1) (7)
(bsp = 1)
𝑘𝑝 ≈ .4𝑘𝑝𝑢 (8)
𝑘𝑖 ≈
𝑘𝑝
.8∗𝑇𝑢
≈ .5 ∗
𝑘𝑝𝑢
𝑇𝑢
(9)

7. 4
Experimental Equipment and Procedure
For the experiment, a DCMCT motor unit (figure 1) was used and Labview was
used to input and adjust the variables to display the results.
Figure 1: DCMCT motor unit
The experiment began by powering up the DCMCT motor unit and downloading
the Labview zip file. After, a small value of kp was as added to make sure that the motor
responded. In order to obtain a zero steady state error, no overshoot, and a 2% settling
time, the value of Ts was set to be less than or equal to .25. The largest voltage the motor
can receive is plus or minus 15 volts, so it is important to know that the motor will cut off
if it exceeds this parameter.
The first experiment performed was the Pure Proportional Control. The
reference signal was set to amplitude of 25 rad/s, a frequency of .6 Hz, and an offset of
50 rad/s. The simulated transfer function was set to K ≈ 18 and τ ≈ 0.085. The filter tc
was set to .03 to eliminate unwanted noise. The integral gain (ki) is set to zero and the set

8. 5
point (bsp) to 1. The kp value was then set to .01 and increase by .01 V*s/rad until a
second order response for the tachometer was reached. Then the kpc theoretical value
had to be calculated and compared to the actual kpc value. The theoretical value was
calculated using equation (1). Then the kp value at critical instability kpu and the Tu
period had to be recorded.
The second experiment performed was the Pure Integral Control. The
proportional gain was set to zero. The integral gain was the swept from 0 V*s/rad to 2.5
V*s/rad. Overshot and steady state error then had to be described as ki was increased.
Then the kic theoretical had to be calculated using equation (2). The theoretical value of
kic was then compared to the actual kic value. The theoretical settling time had to be
calculated using equation (3) and compared to the actual settling time.
The third experiment performed was Proportional and Integral Control with
bsp = 0. The ki and kp gain coefficients were calculated using equations (5) and (6)
respectively. These values were plugged into the program and a response was given.
The Ts estimated value from equation (4) and the actual Ts value were compared. After,
these parameters were adjusted manually in order to achieve a better response.
The fourth experiment performed was Proportional and Integral Control with
bsp = 1. The gain parameters were given from equations (8) and (9). They were input
into the ZN values when running the experiement. The response was plotted and the
overshoot and settling time was recorded.

9. 6
Results
Pure Proportional Control:
The pure proportional control experiment left the ki value at zero in the transfer
function. As the kp value was increased, the RPMs of the wheel increased as well. The
wheel also had a fluctuation in which it would spin fast then slow down repeatedly. At
the beginning the wheel was slightly sticking which gave a different actual response than
the simulated. Using equation (2) gave the kp value equaling .016476. The actual value
from the experiment was 0.06. The first graph (Figure 2) was when kp=kpc(actual). The
second graph (Figure 3) was when kp=kpu which was 0.4 and the Tu value was 3.34
seconds.
Figure 2: Speed as a function of time at kp=kpc(actual)

10. 7
Figure 3: Speed as a function of time at kp=kpu

11. 8
Pure Integral Control:
Pure integral control gave a different response than the pure proportion control.
The kp constant was set to zero in the transfer function. After, the ki value was increased
and a response occurred. At around ki=.1 the wheel accelerates rather quickly, then
slows down. As ki continues to increase more and more, it begins to oscillate back and
forth, faster and faster. The simulated response stays at a constant 15 rad/s. The kic
value and Ts values were calculated using equations (3) and (4). The calculated value of
Kic was .12077 and the actual value was .15 which gave 19.5% error. The Ts value was
calculated to be .92 seconds and the actual value was .87 seconds giving 5.4% error. The
ki actual value was input into the Labview software which gave the following response in
figure 4.
Figure 4: Speed as a function of time at ki=kic(actual)

12. 9
Proportional and Integral Control:
Proportional and integral control is a combination of both proportional and
integral control. First, the response was graphed after calculating ki and kp with bsp
equal to zero. The ki and kp values were calculated using equations (5), (6), and (7). The
ki value was calculated to be 1.635555 and kp was calculated to be .1489. These values
were plugged into the software and are shown in figure 5. Then Ts(actual) was compared
to Ts(theoretical). The theoretical Ts was .25 and the actual Ts was .66. Then the ki and
kp values were manually tuned. The manually tuned values were ki=2.25, kp=.19, and
Ts=.28. These parameters improved the settling time by .38 seconds.
Figure 5: Speed as a function of time at bsp=0, ki=1.6356, Ts=.25, and kp=.1489

13. 10
The bsp value was then set to 1. The values of the two k values were calculated
using the ZN method and observations were made. The kp value was calculated using
equation (8) and the ki value was calculated using equation (9). The estimated
coefficients were somewhat similar but it had a smaller settling time. The kp value was
calculated to be .16 and ki was calculated to be .02395. These values were plugged into
the software and the results are shown in figure 6. The overshoot was calculated to be
1.0983 and the settling time of .07 seconds. There was no saturation because of the large
change in response with variable change. There is more overshoot as the set point
weighting factor increases. The recorded settling time was 0.1.
Figure 6: Speed as a function of time at bsp=1, kp=.16, and ki=.02395

14. 11
Discussion
The function of a DCMCT motor unit is important to understanding control
systems. As demonstrated in the experiment, there is a different response for the pure
proportional control, integral control, and the combined proportional and integral control.
These differences are important to understand when setting the parameters because
different parameters give different responses.
For proportion control, the steady state error was low, along with low overshoot
and minimal settling time for the 2% steady state error. The integral control had the
highest steady state error with no overshoot and a large settling time for the 2% steady
state error. The integral and proportional control with the bsp=0 had a medium steady
state error with low overshoot, and medium settling time for the 2% steady state error.
When the bsp was set to 1 it had the highest steady state error with lots of overshoot and
small settling time for the 2% steady state error.
The optimal values recorded in the experiment to be as ki = 2.25, kp = .19, and Ts
= .28 seconds. These optimal results improved the settling time by .38 seconds. This
was achieved by manually tuning the integral and proportional control method. There
was low steady state error, low overshoot, and a small settling time.

15. 12
Conclusion
In summary, the operation of the DCMCT motor helps give a better
understanding of control systems. Knowing how different inputs changes the response is
important when trying to tune a system. This experiment has many different
applications. The main application is for motorized vehicles. Components such as the
cruise control are assisted using this method. The goal was achieved to choose the kp
and ki values so that there is no steady state error, there is no overshoot, and the 2%
settling time is less than or equal to .25 seconds. It was found to use the manually tuned
parameters in the proportional and integral control method.

16. 13
Reference
Control Systems Pre-lab number 6 for Speed Control